I am working on an example of Hartogs' extension theorem.
Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open subset of $\mathbb{C}^n$, ($n ≥ 2$) and $K$ is a compact subset of $G$. If the complement $G \setminus K$ is connected, then $f$ can be extended to a unique holomorphic function $F$ on $G$.
If we consider the 2D case $f(z,w)= \frac{ 1 }{ z^2 + w^2 + 1 }$, then $f$ is holomorphic in the domain $\{ (z,w) \mid |z|^2 + |w|^2 < 1 \}$. Now if we let $G = \{ (z,w) \mid |z|^2 + |w|^2 < 2 \} $, which is open, and let $K = \{ (z,w) \mid 1 \le |z|^2 + |w|^2 < 2 \}$, which is a compact subset of $G$, then $G\setminus K$ is connected and $f$ is holomorphic on $G\setminus K$. So Hartogs' extension theorem implies that $f$ extends to a holomorphic function $F$ on $G$. I am unable to see what the extension would be. For example, what would be $F(i,0)$, where $f(i,0) = \infty$?